Normal-conducting accelerator magnets Lecture 4: Applied … · 2020. 4. 1. · – for coil and yoke material – required for transient eddy current calculations Mechanical and

Thomas Zickler

CERN

Normal-conducting accelerator magnetsLecture 4: Applied numerical design

Joint Universities Accelerator School

JUAS 2020

Archamps, France, 2. – 4. March 2020

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Lecture 4: Numerical design

Which code shall I use?

Introduction to 2D numerical design

How to evaluate the results

Field analysis

Typical application examples

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Numerical designCommon computer codes: Opera (2D) or Tosca (3D), Poisson, ANSYS, Roxie, Magnus, Magnet, Mermaid, Radia, FEMM, COMSOL, etc…

Technique is iterative

– calculate field generated by a defined geometry

– adjust geometry until desired distribution is achieved

Computing time increases for high accuracy solutions, non-linear problems and time dependent analysis compromise between accuracy and computing time

FEM codes are powerful tools, but be cautious:

– Always check results if they are ‘physical reasonable’

– Use FEM for quantifying, not to qualify3

2D

•2D analysis is often sufficient

•magnetic solvers allow currents only perpendicular to the plane

• fast

3D

•produces large amount of elements

•mesh generation and computation takes significantly longer

•end effects included

•powerful modeller

2D-numerical simulation – Design evaluation – Field analysis – Examples – Summary

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Numerical design process

Design process in 2D (similar in 3D):

Create the model (pre-processor or modeller)

Define boundary conditions, set material properties

Calculations (solver)

Visualize and asses the results (post-processor)

Optimization by adjusting the geometry (manually or optimization code)


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Creating the model

GUI vs. Script

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Model symmetries

Note: one eighth of quadrupole could be used with opposite symmetries defined on horizontal and y = x axis

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Boundary conditions

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Material properties

Permeability:

– either fixed for linear solution

– or permeability curve for non-linear solution

– can be anisotropic

– apply correction for steel packing factor

– pre-defined curves available

Conductivity:

– for coil and yoke material

– required for transient eddy current calculations

Mechanical and thermal properties:

– in case of combined structural or thermal analysis

Current density in the coilsData source: Thyssen/Germany

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Mesh generation

element shape element type element size

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Data processing

• linear: predefined constant permeability for a single calculation

• non-linear: permeability table for iterative calculations

Solution

• static

• steady state (sine function)

• transient (ramp, step, arbitrary function, ...)Solver types

• number of iterations,

• convergence criteria

• precision to be achieved, etc...Solver settings

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Analyzing the results

With the help of the post-processor, field distribution and field quality and be visualized in various forms on the pre-processor model:

– Field lines and colour contours plots of flux, field, and current density

– Graphs showing absolute or relative field distribution

– Homogeneity plots

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Field homogeneity in a dipole

Homogeneity along the x-axis

Homogeneity along GFR boundaryA simple judgment of the field quality can be done by plotting the field homogeneity

1)0,0(

),(

0

y

y

B

yxB

B

B

Homogeneity along the x-axis

Homogeneity along GFR boundary

%01.00

B

B

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Field homogeneity in a dipole

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Saturation and field quality

Field quality can vary with field strength due to saturation

Also very low fields can disturb the field quality significantly

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Field quality in a quadrupole

-1.0E-03

-7.5E-04

-5.0E-04

-2.5E-04

0.0E+00

2.5E-04

5.0E-04

7.5E-04

1.0E-03

-17.5 -12.5 -7.5 -2.5 2.5 7.5 12.5 17.5

Gra

die

nt err

or

x [mm]

Gradient error along the x-axis

CERN proposal V2a P=0.98% R=15 mm I=8.2031 A; tapered poles (OPERA)BINP proposal V2b (18.7.) R=15 mm I=8.2031 A (MERMAID)

-1.0E-04

-7.5E-05

-5.0E-05

-2.5E-05

0.0E+00

2.5E-05

5.0E-05

7.5E-05

1.0E-04

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0

Gra

die

nt

err

or

angle [degree]

Gradient error along the GFR boundary

CERN proposal V2a P=0.98% R=15 mm I=8.2031 A; tapered poles (OPERA)

BINP proposal V2b (18.7.) R=15 mm I=8.2031 A (MERMAID)Gradient homogeneity along the x-axis

Gradient homogeneity along circular GFR

1)0,0('

),(

22

yxB

yxB

%1.0'

'

0

B

B

Field error in a quadrupole

1)0,0('

),('

'

'

0

B

yxB

B

B

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Saturation and field quality

Field quality varies with field strength due to saturation

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Saturation

0

1

2

3

4

5

6

7

8

9

10

0

0.5

1

1.5

2

2.5

0 500 1000 1500 2000 2500 3000

Sa

tura

tio

n [

%]

Flu

x d

en

sit

y [

B]

Current [A]

non-linear

linear

Saturation

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Field analysis

Picking up from lecture 1

and introducing dimensionless normalized multipole coefficients

𝑏𝑛 =𝐵n

𝐵N104 and 𝑎𝑛 =

𝐴𝑛

𝐵N104

with BN being the fundamental field of a magnet: BN (dipole)=B1; BN (quad)=B2; …

we can describe each magnet by its ideal fundamental field and higher order harmonic distortions:

1

41

( ) ( )10

n

Ny x n n

n ref

B zB z iB z b ia

R

18

Harmonic distortionsFundamental field

𝐵𝑦 𝑧 + 𝑖𝐵𝑥 𝑧 =

𝑛=1

∞

𝐵𝑛 + 𝑖𝐴𝑛𝑧

𝑅𝑟𝑒𝑓

𝑛−1


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Field analysis

The field quality of a magnet can be also described by:

• Homogeneity plot: – difference between the actual field B and the ideal field Bid, normalized by the ideal field Bid

– can be expressed by multipole coefficients: for a dipole with

• Harmonic distortion factor Fd :

Note: For good field quality, Fd should be a few units in 10-4

Δ𝐵

𝐵=

𝐵 𝑥, 𝑦 − 𝐵𝑖𝑑 𝑥, 𝑦

𝐵𝑖𝑑 𝑥, 𝑦

2 2

Re Re

1;

( ) ( ) ( )K

d f n ref n f

n n N

F R b R a R

𝐵𝑦,𝑖𝑑 𝑥 = 𝐵1

𝐵𝑦 𝑥 = 𝐵1 +𝐵1

104𝑏2

𝑥

𝑅𝑟𝑒𝑓+ 𝑏3

𝑥

𝑅𝑟𝑒𝑓

2

+ 𝑏4𝑥

𝑅𝑟𝑒𝑓

3

+ ⋯

Δ𝐵

𝐵(𝑥) =

1

104𝑏2

𝑥

𝑅𝑟𝑒𝑓+ 𝑏3

𝑥

𝑅𝑟𝑒𝑓

2

+ 𝑏4𝑥

𝑅𝑟𝑒𝑓

3

+ ⋯

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Field analysis

Multipole errors can be divided into two families:

‚Allowed‘ multipoles are design intrinsic and result from the finite size of the poles

n: order of multipole component

N: order of the fundamental field

m: integer number (m≥1)

fully symmetric dipole

allowed: b3, b5, b7, b9, etc.

non-allowed: all others

fully symmetric quadrupole

allowed: b6, b10, b14, b18, etc.


fully symmetric sextupole

allowed: b9, b15, b21, etc.


‚Non-allowed‘ multipoles result from a violation of symmetry and indicate a fabrication or assembly error

)12( mNn

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Asymmetries

Asymmetries generating ‘non-allowed’ harmonics

n = 2, 4, 6, ... n = 3, 6, 9, ...

n = 4 (neg.) n = 4 (pos.) n = 3 n = 2, 3

These errors can seriously affect machine behaviour and must be controlled!

Comprehensive studies about the influence of manufacturing errors on the field quality have been done by K. Halbach.

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Asymmetry in a C-magnet

• C-magnet: one-fold symmetry

• Since the contribution to the integral in the iron has different path lengths

• Finite (low) permeability will create lower B on the outside of the gap than on the inside

• Generates ‘forbidden’ harmonics with n = 2, 4, 6, ... changing with saturation

• Quadrupole term resulting in a gradient around 0.1% across the pole

.constdlHNI

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‚Shimming‘ (often done by ‘try-and-error’) can improve the field homogeneity

1. Add material on the pole edges: field will rise and then fall2. Remove some material: curve will flatten3. Round off corners: takes away saturation peak on edges4. Pole tapering: reduces pole root saturation -> Rogowsky profile

Pole optimization

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

ΔB

/B

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

ΔB

/B

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

ΔB

/B

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

ΔB

/B

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

ΔB

/B

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

ΔB

/B

Case 1

Case 2

Case 3

Case 4

Case 5

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Rogowsky roll-off

The ‘Rogowsky’ profile provides the maximum rate of increase in gap with a monotonic decrease in flux density at the surface, i.e. no saturation at the pole edges!

The edge profile is shaped according to:

For an optimized pole:

x: pole overhang normalized to the gap a: pole overhang: excess pole beyond the edge of the good field region to reach the required

field uniformity

h: magnet gap

1exp

2 h

xhhy

10-1-2

0

1

2

25.0ln14.020

B

B

h

axoptimized

h

aGFR

24


x

y

h/2

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3D Design

Becomes necessary to study:– the longitudinal field distribution

– end effects in the yoke

– end effects from coils

– magnets where the aperture is large compared to the length

– spacial field distribution

– particle motion in electro-magnetic fields

10 Gauss iso-potential surface

Interference study

3D-field homogeneity25


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Case 1: A material problem

Welding seam on stainless-steel vacuum chamber:

• GFR radius: 30 mm

• Chamber radius: 35 mm

• Welding seam diameter: 1 mm

• Rel. permeability of 316 LN: < 1.001

A small distortion can significantly influence the field quality in the GFR!

μr=1.001 μr=1.1

μr=1.01

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Case 2: An interference problem

Significant attenuation of the corrector field due to the close presence of two quadrupole yokes

Courtesy of A. Vorozhtsov

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Case 3: Mechanical deformation

• Mechanical deformation due to magnetic pressure can influence the field homogeneity

• Multi-physics models can help to quantify the effect

Courtesy of D. Schoerling

Deformation

von Mises stress

Field homogeneity calculated for the center line of the magnet with ANSYS magnetic, ANSYS structural + magnetic, and Opera ST 2D

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Limitations of numerial calculation

Advantages• predict behaviour without having the physical object• for relatively simple cases they are fast and inexpensive

Limitations• multi-physics model: including all couplings (thermal, mechanical) and phenomena

(magnetostriction, magneto-resistivity …) that may be relevant is very complex and expensive

• off-nominal geometry: random assembly errors can dominate field distribution and quality; often, a large number of degrees-of-freedom and the resulting combinatorial explosion makes Monte Carlo prediction costly

• material properties uncertainty : inhomogeneous properties cannot practically be measured throughout volume; even homogeneous materials can be measured only within 2-5% typical accuracy

• numerical errors: e.g. singularities in re-entrant corners, boundary location of open regions may spoil results; special techniques (special corner elements, BEM) require special skills and time

• high cost of detailed 3D models ( x2~3 ); transient simulations increase computing time significantly

Computer simulation targeting

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Summary

• A large varity of FE-codes with different features exist – the right choice depends of the complexity of the problem

• The FE-models shall be as simple as possible and adapted to the problem to reduce computing time

• Numeric computations should be used to quantify, not to qualify

• Benchmarking the results with measurements is a good practice

• Computer simulations have a lot of advantages, but also their limitations

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Documents

Normal-conducting accelerator magnets Lecture 4: Applied … · 2020. 4. 1. · – for coil and yoke material – required for transient eddy current calculations Mechanical and